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\begin{document}
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\title{Homework 2}
\author{Introduction to Robotics\\
ncorrell@colorado.edu}
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\begin{enumerate}
\item You want to find a driving route that minimizes the number of turns between two locations in Boulder.
\begin{enumerate}
\item Would you rather employ a sampling-based or a discrete algorithm such as Dijkstra or A* for doing this and why?
\item Provide a possible cost function that encodes both path length $l$ and the number of turns $t$.
\end{enumerate}
\item Assuming points are sampled uniformly at random in a randomized planning algorithm. Calculate the limiting behaviour of the following ratio   (number of points in tree)/(number of points sampled) as the number of points sampled goes to infinity. Assume the total area $A_{total}$ (including free space and obstacles) and the area of free space $A_{free}$ (not including obstacles) within are known.

\item Assuming a k-d-tree is used as a nearest-neighbour data structure to store a RRT data structure, and points are sampled uniformly at random, calculate the  run-time of inserting a point into a tree of size $N$. Use ``big-Oh'' notation, e.g. $\mathcal{O}(N)$.

\item Why does the bandwidth of a Ultra-sound based distance sensor decreases significantly when increasing its dynamic range, but that of a laser range scanner does not for typical operation?

\item You are designing an autonomous electric car to transport goods on campus. As you are worried about cost, you are thinking about whether to use a laser scanner or an ultra-sound sensor for detecting obstacles. As you drive rather slow, you are required to sense up to 15 meters. The laser scanner you are considering can sense up to this range and has a bandwidth of 10Hz.
\begin{enumerate}
\item Calculate the time it takes until you hear back from the US sensor when detecting an obstacle 15m away. Assume that the robot is not moving at this point. Use $c=300m/s$ for the speed of sound.
\item Calculate the time it takes until you hear back from the laser scanner. Hint: you don't need the speed of light to answer this question.
\end{enumerate}

\item A Global Positioning System sensor provides position estimates within a circle of approximately 3m diameter. Every now and then the satellites on the horizon change and the center of this circle moves elsewhere, approximately staying within a 30 meter radius of the true location of the receiver.
\begin{enumerate}
\item Given the error data above, which value corresponds to accuracy and which to precision?
\item The sensor provides 18000 readings per hour. What is its bandwidth?
\end{enumerate}

\end{enumerate}


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